Cosets and quotients

Concepts

• Kernel of a homomorphism 10 exercisesThe kernel of a group homomorphism φcolon G→ H is the set of elements of G that φ sends to the identity e_H of H.
• Coset 10 exercisesThe left coset of a subgroup H of a group G with representative a∈ G is the set of products of a with the elements of H.
• Index of a subgroup 10 exercisesThe index of a subgroup H of a group G is the number of distinct left cosets of H in G.
• Lagrange's theorem 10 exercisesIf H is a subgroup of a finite group G, the order of H divides the order of G, and the quotient is the index [G:H].
• Normal subgroup 10 exercisesA normal subgroup is a subgroup N of a group G that equals all of its conjugates by elements g∈ G.
• Quotient group 10 exercisesGiven a normal subgroup N of a group G, the quotient group G/N is the set of cosets of N with the operation induced by multiplyin…
• First isomorphism theorem 10 exercisesEvery group homomorphism φ:G→ H induces an isomorphism between the quotient of G by the kernel of φ and the image of φ.
• Direct product of groups 10 exercisesThe direct product G× H of two groups G and H is the set of pairs (g,h), with g∈ G and h∈ H, with the operation defined component…